The original word of 'Arati' is 'ArAtrika'.According to the sanskritdictionary.com, ArAtrika has to be done as follows:
'सर्वेषु चाङ्गेषु च सप्तवारान् आरात्रिकं भक्त- जनस्तु कुर्यात्.: 'sarveshu chAngeshu cha saptavAran ArAtrikam bhakta-janastu kuryAt' meaning that a devotee should do the Arati seven times around all the limbs of the Deity. It also quotes a vachana of Sankaracharya that reads that the devotees doing Arati receives the grace of God : 'शिरसि निहितभारं पात्रमारात्रिकस्य भ्रमयति मयि भूयस्ते कृपार्द्रः कटाक्षः'.
According to the Siddha Gaudiya Vaishnava Sri Krishnaprem (https://en.m.wikipedia.org/wiki/Krishna_Prem), the basic reason behind arati is described as follows :
The whole day we are attracted by objects for pleasing our five sense orgnas - eyes (beauty), tongue(taste), nose (fragrance), skin (touch/contact) and ears (sweet sounds). The arati is just to remind us that all such activities should be done keeping God at the centre .The Circular Movement is the divine movement.
We move dipa (agni-tattva, symbolic of rupa), jala-sankha (jala-tattva, symbolic of taste), dhupa/flowers (prthvi-tattva, symbplic of smell), charama or fan(vayu tattva, symbolic of touch) and ghanta/bell(AakAsha-tattva, symbolic of sounds). The Arati reminds us that we should be aware of the goal of life.
And touching the Arati items after Arati is completed by the devotees seem to signify that we would accept the five tattvas described above only after offering to the Lord and accept His/Her prasada.
Reference : Krishnaprem, Mahajan-Samvad,, Dr. Govindagopal Mukhopadhyay,mPrachi Prakashani, 2nd edition., page 60.